If you want to address concentrated wealth address the tax system and real crony capitalism (not the fake kind that is used to slur Buffett). Eliminate patents, tariffs, tax businesses that receive government subsidies, etc.

Competition is the best bar to too much competition.

While I also long for the elimination of some of the things you list (mainly patents), I want to emphasize that even in a perfectly competitive and ideal market, the phenomenon of preferential wealth accretion would still happen.

Picketty's explanation is that capital's reward is higher than work's, but I'm not bent on defending that explanation (even if until a better one comes up, I'll stick to Picketty's).

As far as I know there is no counter example of the phenomenon itself (whatever the underlying reason is).

You can't stop some people from working harder or smarter than others, or saving money instead of spending everything they make. Those people will accumulate wealth far faster over time.

That is where you are mistaken. The people who will accumulate wealth the fastest over time are the people who already have wealth.

Even statistical anomalies like Bill Gates or Steve Jobs got more money, once they became slightly rich, from the capital they owned than from the work they produced.

Only because they didn't consume their capital, they kept it reinvested, and kept working. There have been thousands of business owners who could have become billionaires, but traded too much of their equity for a big house and expensive distractions.